TITLE: Probability of Selecting A Random Set in Graph with $k$ Edges That is Independent.
QUESTION [7 upvotes]: In a graph with $k$ edges, if we pick every vertex randomly and independently with a probability of $\frac{1}{2}$, prove that
the probability that this set of randomly chosen vertices is an independent set is greater than $\frac{1}{2^k}$.
The approach I am using involves considering all possible selections of vertices, and bounding the probability that those selections have an independent set.
Thanks so much!

REPLY [4 votes]: Choose an arbitrary node from each of the $k$ edges, thereby forming a set of $K \le k$ nodes.  If none of these nodes is selected, the set of selected nodes is independent, because it is missing at least one node from each edge.  The probability that none of the chosen nodes is selected is exactly $2^{-K} \ge 2^{-k}$.